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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"0711cc344752873363f948ece1f7d9fe0ec4f2ed0b215df419f0d786f712d0dd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-180398bbec0732d0","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: coefficients d(n,k) of Andr\u00e9 polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      0;\n  1:      0;\n  2:      1;\n  3:      4;\n  4:     19,      8;\n  5:     94,     68;\n  6:    519,    564,    102;\n  7:   3144,   4512,   1488;\n  8:  20903,  37488,  18816,  1984;\n  9: 151418, 325668, 218976, 44224;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(n^3-5*n^2+6*n-2+(2*n^3-12*n^2+18*n-8)*x+(n^3-7*n^2+14*n-8)*x^2)*P(n-2,x) + (n^2-2*n+1+(4*n^2-9*n+5)*x+(5*n^2-13*n+8)*x^2+(2*n^2-6*n+4)*x^3)*d/dx P(n-2,x) + (2*n^2-7*n+7+(2*n^2-10*n+12)*x)*P(n-1,x) - (n-2+(3*n-6)*x+(2*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 5*n^2 + 6*n - 2 + (2*n^3 - 12*n^2 + 18*n - 8)*x + (n^3 - 7*n^2 + 14*n - 8)*x^2)*P[n - 2, x] + (n^2 - 2*n + 1 + (4*n^2 - 9*n + 5)*x + (5*n^2 - 13*n + 8)*x^2 + (2*n^2 - 6*n + 4)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 7*n + 7 + (2*n^2 - 10*n + 12)*x)*P[n - 1, x] - (n - 2 + (3*n - 6)*x + (2*n - 4)*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(n^3 - 5*n^2 + 6*n - 2 + (2*n^3 - 12*n^2 + 18*n - 8)*x + (n^3 - 7*n^2 + 14*n - 8)*x^2)*v[i - 2] + (n^2 - 2*n + 1 + (4*n^2 - 9*n + 5)*x + (5*n^2 - 13*n + 8)*x^2 + (2*n^2 - 6*n + 4)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 7*n + 7 + (2*n^2 - 10*n + 12)*x)*v[i - 1] - (n - 2 + (3*n - 6)*x + (2*n - 4)*x^2)*deriv(v[i - 1], x))/(n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ff270bcb802a85af","steps":["\u03b3 (scale 1/2) (shift +1)","D","x \u21a6 x+1"],"edges":["e-977fc5754e9861fd","e-4e3b6373024a618e","e-86dfdbb79fe824ab"],"text":"A008292 \u2192 \u03b3 (scale 1/2) (shift +1) \u2192 D \u2192 x \u21a6 x+1"},"headline":"A008292 \u2192 \u03b3 (scale 1/2) (shift +1) \u2192 D \u2192 x \u21a6 x+1","chain":null}